A Multiplication Based Logic Puzzle

Stetson.edu alerted me to the fact that 871 is the smallest number that is the sum of three triangular numbers 23 different ways. I set out to find those 23 ways. We’ll have to count 0 as the 0th triangular number for four of them, but here are those record-breaking 23 ways:

871 is the hypotenuse of Pythagorean triple 335-804-871 which is (5-12-13) times 67.